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Uspekhi Mat. Nauk, 2014, Volume 69, Issue 2(416), Pages 23–76 (Mi umn9575)  

This article is cited in 8 scientific papers (total in 8 papers)

Green's function asymptotics and sharp interpolation inequalities

S. V. Zelika, A. A. Ilyinbc

a University of Surrey, Guildford, UK
b M. V. Keldysh Institute for Applied Mathematics of the Russian Academy of Sciences
c A. A. Kharkevich Institute for Information Transmission Problems of the Russian Academy of Sciences

Abstract: A general method is proposed for finding sharp constants for the embeddings of the Sobolev spaces $H^m(\mathscr{M})$ on an $n$-dimensional Riemannian manifold $\mathscr{M}$ into the space of bounded continuous functions, where $m>n/2$. The method is based on an analysis of the asymptotics with respect to the spectral parameter of the Green's function of an elliptic operator of order $2m$ whose square root has domain determining the norm of the corresponding Sobolev space. The cases of the $n$-dimensional torus $\mathbb{T}^n$ and the $n$-dimensional sphere $\mathbb{S}^n$ are treated in detail, as well as certain manifolds with boundary. In certain cases when $\mathscr{M}$ is compact, multiplicative inequalities with remainder terms of various types are obtained. Inequalities with correction terms for periodic functions imply an improvement for the well-known Carlson inequalities.
Bibliography: 28 titles.

Keywords: Sobolev inequalities, interpolation inequalities, Green's function, sharp constants, Carlson inequality.

Funding Agency Grant Number
Russian Foundation for Basic Research 12-01-00203
11-01-00339
Ministry of Education and Science of the Russian Federation 8502
Russian Academy of Sciences - Federal Agency for Scientific Organizations
This work was carried out with the financial support of the Russian Foundation for Basic Research (grant nos. 12-01-00203 and 11-01-00339), the Russian Ministry of Education and Science (contract no. 8502), and Programme no. 1 of the Russian Academy of Sciences.


DOI: https://doi.org/10.4213/rm9575

Full text: PDF file (978 kB)
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English version:
Russian Mathematical Surveys, 2014, 69:2, 209–260

Bibliographic databases:

UDC: 517.518+517.972
MSC: Primary 46E35; Secondary 35J08, 58J05
Received: 27.10.2013

Citation: S. V. Zelik, A. A. Ilyin, “Green's function asymptotics and sharp interpolation inequalities”, Uspekhi Mat. Nauk, 69:2(416) (2014), 23–76; Russian Math. Surveys, 69:2 (2014), 209–260

Citation in format AMSBIB
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    Citing articles on Google Scholar: Russian citations, English citations
    Related articles on Google Scholar: Russian articles, English articles

    This publication is cited in the following articles:
    1. A. A. Ilyin, “A class of sharp inequalities for periodic functions. Addendum to the paper “Smooth solutions of the Navier-Stokes equations” by S. I. Pokhozhaev”, Sb. Math., 205:2 (2014), 220–223  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib  elib
    2. A. Ilyin, A. Laptev, S. Zelik, “Sharp interpolation inequalities for discrete operators and applications”, Bull. Math. Sci., 5:1 (2015), 19–57  crossref  mathscinet  zmath  isi  elib  scopus
    3. S. V. Zelik, A. A. Ilyin, A. A. Laptev, “Sharp interpolation inequalities for discrete operators”, Dokl. Math., 91:2 (2015), 215–219  crossref  crossref  mathscinet  zmath  isi  elib  elib  scopus
    4. A. A. Ilyin, A. A. Laptev, “Lieb-Thirring inequalities on the torus”, Sb. Math., 207:10 (2016), 1410–1434  mathnet  crossref  crossref  mathscinet  adsnasa  isi  elib
    5. A. A. Ilin, Yu. G. Rykov, “O blizosti traektorii dlya modelnykh kvazigazodinamicheskikh uravnenii. Lineinyi sluchai”, Preprinty IPM im. M. V. Keldysha, 2016, 090, 14 pp.  mathnet  crossref
    6. A. Ilyin, A. Laptev, M. Loss, S. Zelik, “One-Dimensional Interpolation Inequalities, Carlson–Landau Inequalities, and Magnetic Schrödinger Operators”, Int. Math. Res. Notices, 2016, no. 4, 1190–1222  crossref  mathscinet  zmath  isi  elib  scopus
    7. Ilyin A., Laptev A., Zelik S., “Lieb-Thirring Constant on the Sphere and on the Torus”, J. Funct. Anal., 279:12 (2020), 108784  crossref  mathscinet  zmath  isi
    8. S. V. Zelik, A. A. Ilyin, A. G. Kostyanko, “Estimates for the Dimension of Attractors of a Regularized Euler System with Dissipation on the Sphere”, Math. Notes, 111:1 (2022), 47–57  mathnet  crossref  crossref
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